Directional EDA

df <- read_csv('../ocean-waves.csv')

dir.breaks <- seq(0, 360, length.out=9)
dir.labels <- c('N','NE','E','SE','S','SW','W','NW','')

Distribution of direction per location

p1 <- ggplot(df, aes(RottDirection, fill=after_stat(count))) +
    geom_histogram(binwidth=5) + 
    coord_polar() +
    scale_fill_viridis_c(option='plasma') +
    scale_x_continuous(breaks=dir.breaks, labels=dir.labels, limits=c(0,360)) +
    theme(axis.text.y=element_blank(), axis.title=element_blank(),
          axis.ticks.y=element_blank(), legend.position='none',
          panel.grid.minor=element_blank()) +
    ggtitle('Rottnest')

p2 <- ggplot(df, aes(CottDirection, fill=after_stat(count))) +
    geom_histogram(binwidth=5) + 
    coord_polar() +
    scale_fill_viridis_c(option='plasma') +
    scale_x_continuous(breaks=dir.breaks, labels=dir.labels, limits=c(0,360)) +
    theme(axis.text.y=element_blank(), axis.title=element_blank(),
          axis.ticks.y=element_blank(), legend.position='none',
          panel.grid.minor=element_blank()) +
    ggtitle('Cottesloe')

grid.arrange(p1, p2, ncol=2)

p.rot <- ggplot(df, aes(RottDirection, y=RottPeakPeriod, colour=RottHeight)) +
    geom_point(alpha=0.4) + 
    coord_polar() +
    scale_colour_viridis_c(option='plasma') +
    scale_x_continuous(breaks=dir.breaks, labels=dir.labels, limits=c(0,360)) +
    scale_y_continuous(breaks=seq(0, 30, 5), limits=c(0,30)) +
    ylab('Peak period (s)') +
    theme(legend.position='none',
        axis.title.x=element_blank(),
        axis.title.y=element_text(colour='#9f9f9f'),
        axis.text.y=element_text(colour='#9f9f9f'),
        axis.ticks=element_line(colour='#9f9f9f'),
        panel.grid.minor=element_blank()) +
    ggtitle('Rottnest')

p.cot <- ggplot(df, aes(CottDirection, y=CottPeakPeriod, colour=CottHeight)) +
    geom_point(alpha=0.4) + 
    coord_polar() +
    scale_colour_viridis_c(option='plasma') +
    scale_x_continuous(breaks=dir.breaks, labels=dir.labels, limits=c(0,360)) +
    scale_y_continuous(breaks=seq(0, 30, 5), limits=c(0,30)) +
    ylab('Peak period (s)') +
    theme(legend.position='none',
        axis.title.x=element_blank(),
        axis.title.y=element_text(colour='#9f9f9f'),
        axis.text.y=element_text(colour='#9f9f9f'),
        axis.ticks=element_line(colour='#9f9f9f'),
        panel.grid.minor=element_blank()) +
    ggtitle('Cottesloe')

grid.arrange(p.rot, p.cot, ncol=2)

We note some perculiar outliers in Cottesloe, with large swells supposedly coming from the NE (highly unlikely).

Relationship between Rottnest and Cottesloe direction

p.xy.dir <- ggplot(df, aes(RottDirection, CottDirection, colour=abs(RottDirection-CottDirection))) +
    geom_point(alpha=0.5) +
    scale_x_continuous(breaks=seq(0, 360, 90), limits=c(0,360)) +
    scale_y_continuous(breaks=seq(0, 360, 90), limits=c(0,360)) +
    scale_colour_viridis_c(option='plasma') +
    theme(legend.position='none')

p.xy.dir

Total distribution of Direction

all.dir <- df[c('RottDirection','CottDirection')] %>% reshape2::melt()

No id variables; using all as measure variables

all.dir <- all.dir %>% rename(degrees=value)

p.all.dir <- ggplot(all.dir, aes(degrees, fill=after_stat(count))) +
    geom_histogram(binwidth=5) + 
    scale_x_continuous(breaks=dir.breaks, labels=dir.labels, limits=c(0,360)) +
    coord_polar() +
    scale_fill_viridis_c(option='plasma') +
    theme(axis.text.y=element_blank(), axis.title=element_blank(),
          axis.ticks.y=element_blank(), legend.position='none') +
    ggtitle('Rottnest & Cottesloe')

p.all.dir

Searching for suitable angle to rotate axis

(Where there is a lack of data)

dir.bins <- all.dir %>% 
    mutate(bin=cut(degrees, seq(0,360,20), labels=seq(1,360,20)),
           bin=as.numeric(as.character(bin))-1) %>%
    group_by(bin) %>%
    summarise(freq=n())

dir.p1 <- ggplot(all.dir, aes(degrees, fill=after_stat(count))) +
    geom_histogram(binwidth=10) +
    geom_vline(aes(xintercept=40), colour='tomato', lwd=1) +
    scale_x_continuous(breaks=dir.breaks, limits=c(0,360)) +
    ylim(c(0,100)) +
    coord_polar() +
    scale_fill_viridis_c(option='plasma') +
    theme(axis.text.y=element_blank(), axis.title=element_blank(),
          axis.ticks.y=element_blank(), legend.position='none')

dir.p2 <- ggplot(all.dir, aes(degrees, fill=after_stat(count))) +
    geom_histogram(binwidth=10) +
    geom_vline(aes(xintercept=40), colour='tomato', lwd=1) +
    coord_cartesian(ylim=c(0,100)) +
    scale_fill_viridis_c(option='plasma') +
    theme(axis.text.y=element_blank(), axis.title=element_blank(),
          axis.ticks.y=element_blank(), legend.position='none') +
    ggtitle('Least common swell directions')

grid.arrange(dir.p2, dir.p1, ncol=2)

40 degrees is found to be a suitable candidate for axis rotation.

rotate.degrees <- function(deg, rotate) {
    deg <- deg + rotate
    if (deg >= 360) {
        deg <- deg - 360
    } else if (deg < 0) {
        deg <- deg + 360
    }
    return (deg)
}

all.dir$rotated <- all.dir$degrees %>% sapply(rotate.degrees, rotate=-40)

p.all.dir.rotated <- ggplot(all.dir, aes(rotated, fill=after_stat(count))) +
    geom_histogram(binwidth=5) + 
    scale_x_continuous(breaks=dir.breaks, limits=c(0,360)) +
    geom_vline(aes(xintercept=0), colour='tomato', lwd=1) +
    coord_polar() +
    coord_polar(start=40*pi/180) +
    scale_fill_viridis_c(option='plasma') +
    theme(axis.text.y=element_blank(), axis.title=element_blank(),
          axis.ticks.y=element_blank(), legend.position='none') +
    ggtitle('Rottnest & Cottesloe, rotated -40°')

Coordinate system already present. Adding new coordinate system, which will replace the existing one.

grid.arrange(
    p.all.dir +
        scale_x_continuous(breaks=dir.breaks, limits=c(0,360)) +
        geom_vline(aes(xintercept=0), colour='tomato', lwd=1),
    p.all.dir.rotated,
    ncol=2
)

p.xy.dir.rotated <- ggplot(df, aes(
        x=RottDirection %>% sapply(rotate.degrees, rotate=-40),
        y=CottDirection %>% sapply(rotate.degrees, rotate=-40),
        colour=abs(RottDirection-CottDirection))) +
    geom_point(alpha=0.5) +
    scale_colour_viridis_c(option='plasma') +
    theme(legend.position='none') +
    labs(x='Rottnest Direction rotated -40°',
         y='Cottesloe Direction rotated -40°',
         title='After -40° rotation')

grid.arrange(p.xy.dir + ggtitle('Before rotation'), p.xy.dir.rotated, ncol=2)

with(df, abs(RottDirection - CottDirection) %>% mean())

[1] 18.90977

with(df, abs(
    sapply(RottDirection, rotate.degrees, rotate=-40) -
    sapply(CottDirection, rotate.degrees, rotate=-40)) %>% mean())

[1] 18.84041

Convert to radians

[-pi, pi)

all.dir$radians <- (all.dir$degrees * pi / 180) - pi
all.dir$rotated.radians <- (all.dir$rotated * pi / 180) - pi

radian.grid <- seq(-pi, pi, length.out=5) %>% round(2)
radian.labels <- c('-π', '-π/2', '0', 'π/2', 'π')
radian.labels.polar <- c('', '-π/2', '0', 'π/2', 'π, -π')

p.all.dir.radians <- ggplot(all.dir, aes(radians, fill=after_stat(count))) +
    geom_histogram(binwidth=2*pi/(360/5)) + 
    scale_x_continuous(breaks=radian.grid, labels=radian.labels.polar, limits=c(-pi, pi)) +
    geom_vline(aes(xintercept=pi), colour='darkgrey', lwd=1) +
    coord_polar() + scale_fill_viridis_c(option='plasma') +
    theme(axis.text.y=element_blank(), axis.title=element_blank(),
          axis.ticks.y=element_blank(), legend.position='none') +
    ggtitle('Radians, no rotation')

p.all.dir.rotated.radians <- ggplot(all.dir, aes(rotated.radians, fill=after_stat(count))) +
    geom_histogram(binwidth=2*pi/(360/5)) + 
    scale_x_continuous(breaks=radian.grid, labels=radian.labels.polar, limits=c(-pi, pi)) +
    geom_vline(aes(xintercept=pi), colour='darkgrey', lwd=1) +
    coord_polar(start=40*pi/180) + scale_fill_viridis_c(option='plasma') +
    theme(axis.text.y=element_blank(), axis.title=element_blank(),
          axis.ticks.y=element_blank(), legend.position='none') +
    ggtitle('Radians, after -40° rotation')

grid.arrange(p.all.dir.radians, p.all.dir.rotated.radians, ncol=2)

df.dir.processed <- df %>% mutate(
    CottRadians=(CottDirection * pi / 180) - pi,
    RottRadians=(RottDirection * pi / 180) - pi,
    CottRadiansRotated=(sapply(CottDirection, rotate.degrees, rotate=-40) * pi / 180) - pi,
    RottRadiansRotated=(sapply(RottDirection, rotate.degrees, rotate=-40) * pi / 180) - pi,
)

---

Projecting Direction onto the unit circle

p.cott.circ3 <- ggplot(df.dir.processed, aes(sin(pi*CottDirection/180), cos(pi*CottDirection/180))) +
    geom_point(colour='#d34f72', alpha=0.3, size=2.5) +
    ggtitle('Cottesloe') + labs(x='sin(Cottesloe Radians)', y='cos(Cottesloe Radians)')
p.rott.circ3 <- ggplot(df.dir.processed, aes(sin(pi*RottDirection / 180), cos(pi*RottDirection/180))) +
    geom_point(colour='#5012a0', alpha=0.3, size=2.5) +
    ggtitle('Rottnest') + labs(x='sin(Rottnest Radians)', y='cos(Rottnest Radians)')
grid.arrange(p.cott.circ3, p.rott.circ3, ncol=2)

df.dir.processed <- df.dir.processed %>% 
    mutate(
        sinCottRad=sin(CottRadians),
        cosCottRad=cos(CottRadians),
        sinRottRad=sin(RottRadians),
        cosRottRad=cos(RottRadians)
    )

write_csv(df.dir.processed, '../ocean-waves-dir-processed.csv')